Robustly Optimal Auctionswith Unknown Resale Opportunities�

Gabriel Carroll Ilya Segal

Department of EconomicsStanford University, Stanford, CA 94305, USA

February 13, 2018

Abstract

The standard revenue-maximizing auction discriminates against apriori stronger bidders so as to reduce their information rents. Weshow that such discrimination is no longer optimal when the auctionswinner may resell to another bidder, and the auctioneer has non-Bayesian uncertainty about such resale opportunities. We identify aworst-caseresale scenario, in which biddersvalues become publiclyknown after the auction and losing bidders compete Bertrand-style tobuy the object from the winner. With this form of resale, misalloca-tion no longer reduces the information rents of the high-value bidder,as he could still secure the same rents by buying the object in resale.Under regularity assumptions, we show that revenue is maximized bya version of the Vickrey auction with bidder-specic reserve prices,

�We are grateful to Adrien Auclert, Lawrence M. Ausubel, Ben Brooks, Peter Cramton,Drew Fudenberg, Simon Grant, Paul Milgrom, Isaac Sorkin, Glen Weyl, and participantsof the 2016 Stony Brook Workshop on Complex Auctions and Practice, the NBER Mar-ket Design Working Group, the NYUAD Advances in Mechanism Design conference, theNSF-CEME Decentralization conference, and seminars at ANU, UNSW, UQ, HU-Berlin,Mannheim, Bonn, Northwestern, and Harvard-MIT, for helpful comments. We thank theSimons Institute at UC Berkeley for its hospitality during the Fall 2015 Economics andComputation Program. Segal acknowledges the support of the National Science Founda-tion (grant SES 0961693). Carroll is supported by a Sloan Research Fellowship. AlexBloedel and Seunghwan Lim provided valuable research assistance.

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rst proposed by Ausubel and Cramton (2004). The proof of optimal-ity involves constructing Lagrange multipliers on a double continuumof binding non-local incentive constraints.

1 Introduction

Standard auction theory says that when bidders are a priori asymmetric,revenue-maximizing auctions discriminate against stronger bidders, i.e.those who are more likely to have higher values. The optimal auction re-quires them to pay a premium over weakerbiddersvalues in order to win(Myerson 1981, McAfee and McMillan 1989). This discrimination enhancesrevenue because it increases competition between weaker and stronger bid-ders and so reduces the latter biddersinformation rents.One might suspect that the benets of such discrimination would be vi-

tiated if bidders could resell to each other after the auction, since a strongbidder might then prefer to sit back and let a weaker bidder win, in thehopes of buying from him later at a better price. In the words of Ausubeland Cramton (1999, p. 19): The possibility of resale undermines the sellersability to gain by misassigning the good. The best that the seller can dois to conduct an e¢ cient auction [i.e. selling only to the highest-value bid-der], perhaps withholding ... the good.1 However, this is not supported byexisting results on optimal auctions with resale (Zheng 2002, Calzolari andPavan 2006), in which revenue is typically maximized by a biased auction,which induces resale in equilibrium with a positive probability. For a simpleexample due to Zheng (2002), suppose one bidder is known to have a zerovalue for the good and also to have full bargaining power in resale. Thenthe auctioneer would want to simply sell to this bidder, who would then re-sell the good using Myersons (1981) revenue-maximizing mechanism. Theauctioneer could charge the zero-value bidder a price equal to the Myersonexpected revenue. (This is the best she can do as long as biddersvaluesremain private information, since the combined outcome of the auction andresale would have to satisfy the same incentive compatibility constraints as

1The notion that price discrimination is ine¤ective under resale is also familiar in thecontext of pricing in markets. For example, Tirole (1988, p. 134) writes: It is clear thatif the transaction (arbitrage) costs between two consumers are low, any attempt to sell agiven good to two consumers at di¤erent prices runs into the problem that the low-priceconsumer buys the good to resell it to the high-price one.

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Myersons mechanism.)Ausubel and Cramton (1999, 2004) claim that it is optimal for the auc-

tioneer not to misallocate when resale is perfect.Specically, they proposea way of adding bidder-specic reserve prices to a Vickrey (second-price)auction, which we call the Ausubel-Cramton-Vickrey (ACV) auction anddescribe in detail below. The ACV either allocates the object e¢ cientlyamong bidders, or withholds it (when no bidder beats his assigned reserve).2

Ausubel and Cramton (1999) assert that the ACV auction is optimal. Theydo not formalize the assumption of perfect resale, and as is well known, ef-cient resale would generally be impossible under private information;3 butone might justify the assumption by assuming that the partiesvalues exoge-nously become publicly known before resale.However, even if we operationalize perfect resale in this way, it could

still be optimal for the auctioneer to misallocate. Indeed, if we modify thezero-value bidder example above by letting the zero bidder observe all otherbiddersvalues before reselling, the optimal mechanism would extract fullrst-best surplus by again selling the object to the zero bidder, now at aprice equal to the expectation of the highest value, which is exactly what thezero bidder expects to receive in resale. Similarly, in the symmetric settingwith perfect resale studied by Bulow and Klemperer (2002) and Bergemannet al. (2017a), where the auctions winner is assumed to have full informationand full bargaining power in resale, the auctioneer can command a higherprice by allocating to an ine¢ cient bidder than to the e¢ cient one, becausethe ine¢ cient bidder can later extract the e¢ cient bidders surplus in resale;thus the optimal auction misallocates the good.4

2At least two other ways to introduce asymmetric reserves into Vickrey have been stud-ied: lazyand eagerVickrey (Dhangwatnotai et al., 2010). EagerVickrey allocatesto the highest bidder among those who met their reserve prices as long as one such bidderexists, and so it misallocates in equilibrium. LazyVickrey allocates to the highest bid-der provided that he met his reserve price and does not allocate otherwise. It will emergeas the solution to our relaxed problem in Subsection 4.2.

3For example, when the auction simply gives the object to bidder 1 for zero payment,so that no information is revealed, resale must be ine¢ cient according to the theorem ofMyerson and Satterthwaite (1983). The same ideas apply to resale following more generalauction mechanisms.

4Bulow and Klemperer (2002, Example 3) nd that in this setting, selling to a randomlychosen bidder at a xed price low enough to guarantee a sale yields a higher revenue thanthe Vickrey auction. Bergemann et al. (2017a) show that the xed-price mechanism is infact the optimal mechanism among those that always sell the good. They also derive the

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Our paper shows how the intuition expressed by Ausubel and Cramtoncan nevertheless be validated when the auction designer is uncertain aboutthe resale procedure (including possible exogenous revelation of private in-formation) and desires the revenue to be robust to this uncertainty. Indeed,notice that some of the optimal biased auctions described in the above ex-amples are not robust to even a small amount of uncertainty about resale.5

Thus, we formulate the problem of maximizing worst-case expected revenue,where the expectation is taken over buyersprivate values drawn from knowndistributions, and the worst case is over the possible resale procedures. Werefer to this problem as the robust revenue maximization problem.To solve this problem, we begin with a simplied model in which the auc-

tioneer is required to always sell the object (Section 3). With no resale, theoptimal auction in this model would be biased. In contrast, robust revenuemaximization under resale uncertainty is achieved by the simple second-priceauction (with no reserve), which allocates e¢ ciently. To prove this, we guessa worst-caseresale procedure, in which biddersvalues exogenously becomepublic knowledge after the auction, and then the losing bidders compete àla Bertrand to buy the object from the winner in resale. With this resaleprocedure, in any auction that always sells the object, each bidder can guar-antee himself a payo¤ equal to his marginal contribution to social surplus,by sitting out to let another bidder win and then buying from the winner.Given this lower bound on the information rents captured by the bidders,the designer cannot do better for herself than the Vickrey auction. Since thisauction also sustains truthful bidding as an ex post equilibrium under anyother resale procedure, it solves the robust revenue maximization problem.As a side benet, the optimal auction turns out to be independent of thebiddersvalue distributions, i.e., completely prior-free.While the must-sell model cleanly illustrates the robust optimality of

e¢ cient auctions, in most real-life settings (and in the classical theory) theauctioneer has the ability to increase revenue by sometimes withholding theobject, e.g. by using reserve prices. In Section 4, we turn to study suchsettings, and show that under appropriate regularity assumptions on the

optimal auction that can withhold the good, and show that it misallocates with a highprobability. While written independently of our paper, their paper shares some technicalsimilarities with our analysis, which we point out below.

5In either of the zero-bidder examples, suppose the auctioneers guess about resale ismistaken, and there is actually an " probability that no resale opportunity will arise. Thenthe zero bidder refuses to buy at the proposed price, and revenue drops to zero.

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distributions, an ACV auction with appropriately chosen reserve prices isoptimal. Thus, the auctioneer again never misallocates the good, but takesadvantage of the known asymmetries by setting di¤erent reserve prices todi¤erent bidders. We show this result by proving that the ACV auction isoptimal under the particular worst-case resale procedure described above.Since, by an argument of Ausubel and Cramton (2004) (which we includein the appendix for completeness), this auction sustains truthful biddingas an equilibrium under any resale procedure, it solves the robust revenuemaximization problem.Characterizing an optimal auction under our worst-case resale procedure

requires new techniques that may be of separate interest. First, by foldingthe outcome of resale into reduced-form payo¤ functions, we model the auc-tioneers problem as a single-stage auction design problem, albeit one withexternalities and interdependent values (since a bidder who does not winthe object cares whether another bidder wins, and what the winners valueis.)6 We can then apply the standard rst-order approachto such prob-lems, which considers only the local incentive constraints, and rewrites theobjective as an appropriately-dened virtual surplus.The solution found by this standard method reinforces our basic intu-

ition: it never misallocates the good; it either allocates e¢ ciently or with-holds the good. However, this solution is not the right answer, because it isnot incentive-compatible. It is vulnerable to non-local deviations, where abidder underbids to lose and then buys the good in resale. To nd the trueoptimal auction, we need to account for such non-local incentive constraints.We guess that the optimum is the ACV auction. To verify optimality, we ex-plicitly construct supporting Lagrange multipliers on the double continuumof binding non-local incentive constraints.7

Our approach also yields an iterative construction of the optimal bidder-specic reserve prices in ACV auctions. Appendix F illustrates by working

6Other examples of externalities and interdependent values caused by post-auctioninteractions among buyers can be found in Jehiel and Moldovanu (2000) and Bulow andKlemperer (2002).

7Earlier work that constructed Lagrange multipliers as a measure over a double con-tinuum of binding incentive constraints includes the transport theoryapproach to mul-tidimensional screening (e.g., Daskalakis et al. (2013)). Also, Bergemann, Brooks, andMorris (2017a) independently develop a treatment of non-local incentive constraints thatis the closest technically to our approach: our analysis implicitly shares with theirs thefeature of considering randomized misreports that are drawn from the same distributionas the true type, but truncated.

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through an example in which biddersvalues are distributed uniformly withdi¤erent upper limits. In this case, with bidders ordered from stronger toweaker, the optimal reserve price for the kth bidder is obtained by solving akth-degree polynomial equation.Some readers may nd a tension between the auctioneers use of Bayesian

priors to construct optimal reserve prices in the ACV auction and her com-plete ignorance of the resale procedure. We do believe that there are real-life situations in which the auctioneer has knowledge that some bidders arestronger than others, which is traditionally captured by means of Bayesianpriors, while being ignorant about the resale procedure.8 Furthermore, as wedetail in Section 5, the tension can be directly ameliorated in two ways. First,we o¤er a model with uncertainty both about value distributions and aboutresale, in which our conclusion about optimality of ACV auctions carriesover. Second, we o¤er additional results suggesting that setting the right re-serve prices is quantitatively less important than ignorance of resale, so thatsimply using Vickrey with no reserves is a good prior-free auction choice.Our paper joins the growing literature using the maxmin criterion to

model robust mechanism design (including, in particular, Frankel (2014),Carroll (2015), and Bergemann et al. (2017b)). Conceptually, it is theclosest to the work showing that a strategy-proof mechanism may emergeas optimal when the designer is uncertain about biddersbeliefs about eachothers values (Chung and Ely 2007; Yamashita and Zhu 2017) or about eachothers strategies (Yamashita 2015). Likewise, in our paper, a resale-proofmechanism may emerge as optimal when the designer is ignorant about theresale procedure. Also, observe that ACV auctions are also robust to biddersbeliefs and information about each others values and their beliefs about theresale procedure, hence we obtain these additional robustness benets forfree.

8To give one example, in spectrum auctions run by the U.S. Federal CommunicationsCommission, some bidders (such as AT&T, Verizon, and T-Mobile) are believed to havestrong business cases for the use of additional spectrum, while other bidders (such as Dishand Comcast) are believed to be bidding speculatively, with their values determined to alarge extent by expectation of resale, even though the structure and outcomes of the resalemarket are hard to anticipate.

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2 Setup

There are n � 2 bidders. Bidder is private type is his value �i for the ob-ject, which is distributed on [0; 1] according to a c.d.f. Fi with a continuousstrictly positive density fi.9 Values are independent across bidders. We write� = (�1; : : : ; �n) for the prole of values. The space of (possibly randomized)allocations is X = fx 2 [0; 1]n :

Pi xi � 1g, where xi 2 [0; 1] is the proba-

bility of allocating the object to bidder i. We use tildes to denote randomvariables: ~�i for the random variable representing is value; �i for a specicrealization.A general auction mechanism is a triple � = hM;�; i, where

� M = (M1; : : : ;Mn) is a collection of measurable message spaces forthe bidders, such that ; 2 Mi for each i, where ; denotes the specialnon-participation message;

� � :QiMi ! X is a measurable allocation rule and :

QiMi ! Rn is a

measurable payment rule, with �i (m) = i (m) = 0 whenever mi = ;.

Following allocation specied by the auction, resale may take place. Wemodel resale in reduced form by an n-tuple of measurable functions v =(v1; : : : ; vn), where vi(x; �) gives bidder is post-resale payo¤(net of paymentsin the auction) following allocation x 2 X specied by the auction when thebiddersvalue prole is �. This formalism captures a setting in which allbidders values exogenously become public after the auction, so that theoutcome of resale depends only on the initial allocation and the values. (InAppendix C, we describe a more general class of resale procedures that doesnot assume values are revealed.)We require that the total reduced-form payo¤s not exceed the maximal

total surplus available in resale:Pi vi (x;�) �

�maxi�i

�� (P

i xi) : (1)

We also require the resale procedure to be individually rational:

vi(x; �) � �ixi for each i: (2)9The assumptions of common support and continuous positive densities are made for

expositional simplicity: in Appendix F we explain how our results can be extended tosome cases in which these assumptions do not hold.

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A given mechanism � and resale procedure v together induce a Bayesiangame: the action space of player i is Mi, and his payo¤ is vi (� (m) ; �) � i (m), with corresponding revenue

Pi i(m) for the auctioneer.

Let Rev(�; v) denote the supremum of expected revenue over all Bayes-Nash equilibria of this game. We state the robust revenue maximizationproblem as10

max�

�infvRev(�; v)

�: (3)

We will establish that a specic auction �� solves the robust revenue max-imization problem, by constructing a resale procedure v and a revenue target�R such that the following two conditions are satised:

Rev(�; v) � �R for all auctions �, (4)Rev(��; v) � �R for all resale procedures v: (5)

(4) means that given resale procedure v, the designer could not design an auc-tion yielding expected revenue above �R, while (5) means that given auction��, an adversary could not construct a resale procedure to reduce the de-signers expected revenue below �R. The logic of the Minimax Theorem thenimplies (the formal proof of this and all other results are in the appendix)

Lemma 1. If (4)-(5) hold then auction �� solves the robust revenue maxi-mization problem max� (infv Rev(�; v)), while resale procedure v solves theworst-case resale problem minv (sup�Rev(�; v)), and the value of both prob-lems is Rev(��; v) = �R.

To establish (4) for a particular resale procedure v, we can apply the Rev-elation Principle and restrict attention to direct mechanisms �, in which eachbidder is message space is Mi = [0; 1] [ f;g, and to the Bayes-Nash equi-librium in which all bidders participate and report truthfully, i.e., � satises

10Note, in particular, that Rev(�; v) = �1 if the induced game has no equilibrium,so the designer must guarantee equilibrium existence for all v. We view this requirementas not too onerous; for example, any mechanism � with nite message spaces guaranteesequilibrium existence (Milgrom and Weber, 1985, Theorem 1). It is in line with thestandard assumption in mechanism design that agents must play an equilibrium.

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the following incentive compatibility and individual rationality constraints:11

E~��i [vi(�(�i; ~��i); �i; ~��i)� i(�i; ~��i)] � E~��i [vi(�(�̂i; ~��i); �i; ~��i)� i(�̂i; ~��i)] for all �i; �̂i;(6)

E~��i [vi(�(�i; ~��i); �i; ~��i)� i(�i; ~��i)] � E~��ihvi(�(;; ~��i); �i; ~��i)

i: for all �i:

(7)

Then, we show (4) by showing that the expected revenue in any directauction satisfying (6)-(7) cannot exceed the target �R, which we take to bethe expected revenue of �� when bidders behave truthfully. In Section 3,we do this for the special case in which the seller must sell the object withprobability 1, and �� is the standard Vickrey auction. In Section 4, we dothis for the general case in which the seller can withhold the object, and �� isan ACV auction (formally dened in Denition 1 ahead) with appropriatelyconstructed bidder-specic reserve prices. These results constitute the heartof our contribution. Then, by an argument of Ausubel and Cramton (2004),truthtelling is an equilibrium of �� for any resale procedure v, not just forv. Thus, Rev(��; v), which is the highest expected revenue of �� across allequilibria, is bounded below by �R, establishing (5). Lemma 1 then impliesour main result.

3 The Must-Sell Case

We begin by considering a simpler model in which the seller must sell theobject. (For example, this could be microfounded by assuming the seller hasa prohibitively high cost of keeping the object.) To adapt the robust problem(3) to this setting, just redene the objective Rev(�; v) as the supremum ofexpected revenue over those Bayes-Nash equilibria in which the object is soldwith probability 1, and Rev(�; v) = �1 if no such equilibrium exists.In the absence of resale, if biddersvalues are drawn from di¤erent distrib-

utions, the optimal must-sell auction misallocates the object towards weakerbidders, with the goal of reducing the information rents of stronger bidders

11Note that the Revelation Principle would not generally apply to the robust maximiza-tion problem, since in a given auction, truth-telling may be an equilibrium for some resaleprocedures but not for others. This is why the robust problem is formulated using indirectmechanisms.

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(Myerson (1981), McAfee and McMillan (1989)). As we shall see, this benetof misallocation can be hampered by resale.We guess a worst-caseresale procedure to be the Bertrand gamein

which the auctions losers make competing price o¤ers to acquire the objectfrom the winner, who then accepts one of the o¤ers or rejects all of them andkeeps the object to himself. As a result, if the auctions winner is not thehighest-value bidder, the former sells the object to the latter at price �(2), thesecond-highest value of all bidders.12 Thus, bidder is post-resale payo¤ fromauction allocation x in state � (exclusive of payments made in the auction)can be written as

vi(x; �) = maxf�i; �(2)g � xi +maxf0; �i � �(2)g �P

j 6=i xj: (8)

Note that this resale procedure v is e¢ cient (i.e., satises (1) with equality)and individually rational (satises (2)).Resale procedure (8) is a natural guess for the worst case because it

makes the highest-value bidder a residual claimant for surplus, thus allowinghim to capture information rents even if the object is allocated to anotherbidder. More specically, in any must-sell auction, any bidder i can, byletting another bidder win and then buying from him if possible, assurehimself an expected payo¤ E~�

hmaxf~�i � ~�(2); 0g

i. This payo¤ is bidder is

expected marginal contribution(that is, the total surplus available, minusthe surplus that would be achievable if i were absent), and coincides withhis expected payo¤ in the Vickrey auction (i.e. second-price auction) with noreserve price. If the seller cannot avoid conceding at least this much surplusto the bidders, she cannot do any better than using the Vickrey auction withno reserve price.The above argument is not complete because it is not clear that bidder i

can ensure that another bidder wins while avoiding paying anything to theauctioneer. For example, if bidder i bids exactly 0, the auctioneer mightwithhold the good (since she only needs to sell with probability 1), or might

12Specically, this is the unique outcome arising in a subgame-perfect equilibrium inundominated strategies in the full-information Bertrand game. This is not the only worst-case resale procedure: for example, the argument below would also work if the highest-value bidder, when he fails to win the object, were to buy it back by making a take-it-or-leave-it o¤er to the auctions winner. On the other hand, the results below do rely onthe extreme division of bargaining power in these games. For example, if there are twobidders and bidder 1 captures a xed share � 2 (0; 1) of resale surplus, we can have anexample where the seller can do better than the Vickrey auction.

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sell it to bidder i but charge him a positive price, letting him recoup it inresale. We can address these issues by noting that since the auction must sellwith probability 1, bidder i could ensure that the good is sold with probability1 (to either another bidder or himself) by making a bid �̂i that is arbitrarilysmall. Then, we can show that the individual rationality constraint of type�̂i and resale procedure (8) guarantee bidder i an expected payo¤ close to hisexpected Vickrey payo¤, which implies the result.

Theorem 1. Under resale procedure (8), equilibrium expected revenue in anyauction that sells with probability 1 does not exceed the expected truthtellingrevenue in the Vickrey auction.

This establishes condition (4) above for resale procedure (8). As forcondition (5), it follows from the observation that truthful bidding is an expost equilibrium of the Vickrey auction for any resale procedure satisfying(1)-(2) (and then no resale occurs, since the highest-value bidder has alreadywon). While this will follow from Theorem 2 below, the informal argumentis as follows. Since the usual arguments show that a bidder can never benetby deviating from truthful bidding in a way that does not involve resale, wejust need to check that there are no protable deviations that involve resaleeither. If bidder i makes a downward deviation that causes him to lose, hecannot buy the object in resale unless he pays at least the winners value, �(2),which is what he would have paid to win the object in the auction anyway.And if he makes an upward deviation that causes him to win, he cannot resellthe object for more than the highest value, �(1), which is the price he wouldhave to pay to win the auction. In both cases he does no better than biddinghis true value.With (4) and (5) thus established, Lemma 1 implies

Corollary 1. The Vickrey auction solves the robust revenue maximizationproblem (3) for a seller that must sell the object with probability 1.

4 The Can-Keep Case

We now turn to the optimal auction when the seller can withhold the object,which is the version of the model more widely studied in the literature. Asin the previous section, we conjecture that a worst-case resale procedure isgiven by (8), and show that given this procedure, an appropriately-designedACV auction is optimal.

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4.1 Regularity Assumptions on Distributions

The main result of this section will use the following assumptions on biddersvalue distributions. (However, many of the intermediate steps will not relyon all of the assumptions, and so we will impose them only when needed.)

A1. For each bidder i, the hazard rate fi (�) = [1� Fi (�)] is nondecreasing.

A2. For each bidder i, the reverse hazard rate fi (�) =Fi (�) is nonincreasing.

A3. The virtual value functions �i(�) = �� (1�Fi(�))=fi(�) satisfy �1 (�) �: : : � �n (�) for each � 2 [0; 1].

(A1) implies, in particular, that each distribution Fi is regular, i.e., eachfunction �i(�) is increasing. The value ri = ��1i (0) is then uniquely dened;it is the optimal price for selling to bidder i alone, and is also the optimalreserve price for bidder i in Myersons optimal auction. Note also that both(A1) and (A2) are satised, in particular, when the density fi is log-concave,which is satised by many standard distributions (Bagnoli and Bergstrom2005).Assumption (A3) is equivalent to requiring that the distributions Fi be

ordered from strongerto weakerunder the hazard rate ordering. In theabsence of resale, it is a su¢ cient condition (and, for two-bidder must-sellauctions, a necessary condition) for the optimal auction to always discrim-inate against the stronger bidders (McAfee and McMillan 1989, Theorem3). In particular, (A3) ensures that the optimal bidder-specic reserve priceswould satisfy r1 � : : : � rn.While assumptions (A1)-(A3) are not unreasonable and are familiar from

the literature on optimal auctions, they are not the weakest possible to ensurethe optimality of ACV auctions. In Section 5 we discuss the possibility ofrelaxing the assumptions and the challenges that arise.

4.2 Virtual Surplus and the Relaxed Problem

We begin with the standard approach of using rst-order incentive compat-ibility constraints to substitute out the payment functions. To this end, forany proposed (direct) auction (�; ), let

Ui (�i) = E~��i [vi(�(�i; ~��i); �i; ~��i)� i(�i; ~��i)] (9)

12

be the interim expected payo¤ enjoyed by i when his type is �i, and notethat by the standard envelope-theorem argument, incentive compatibility(6) implies that Ui is absolutely continuous and its derivative is given almosteverywhere by13

U 0i(�i) = E~��i [v0i(�(�i;

~��i); �i; ~��i)]: (10)

Here v0i denotes the derivative of vi(x; �) with respect to �i, which is denedwhen �i does not tie with another bidder, and then takes the form v0i(x; �) =1�i=�(2) � xi + 1�i=�(1).As for participation constraints, as usual we consider (7) only for type 0,

for which it takes the form Ui(0) � 0. If only this participation constraintis imposed, it will be optimal to satisfy it with equality, and so integrating(10) and using (9) yields the following expression for the interim expectedpayment of bidder i of type �i:

E~��i [ i(�i; ~��i)] = E~��i [vi(�(�i; ~��i); �i; ~��i)]�Z �i0

E~��ihv0i(�(�̂i;

~��i); �̂i; ~��i)id�̂i:

(11)The usual integration by parts then allows us to rewrite the auctions

expected revenue in terms of the allocation rule as

E~�

"Xi

i(~�)

#= E~�

"Xi

vi(�(~�); ~�)�

Xi

1� Fi(~�i)fi(~�i)

v0i(�(~�); ~�)

#: (12)

The standard relaxed problem maximizes (12) over all allocation rules�, ignoring (6), as well as (7) for types other than 0. This can be doneby maximizing the virtual surplus (the expression inside brackets) for eachprole � separately. At any prole � without ties, if we allocate to thehighest-value bidder i�, the virtual surplus is

�i� �1� Fi�(�i�)fi�(�i�)

= �i�(�i�):

13Specically, for any given report �̂i, vi(�(�̂i; ��i); �i; ��i) is Lipschitz continuous in�i and is di¤erentiable in �i except when there are ties in values. This implies thatE~��i [vi(�(�̂i;

~��i); �i; ~��i)] is Lipschitz continuous and di¤erentiable in �i, and allows theapplication of Milgrom and Segals (2002) Corollary 1 to establish absolute continuity ofUi and (10).

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If we allocate to the second-highest bidder j, then the v0i terms are 1 for bothi = i� and i = j, so the virtual surplus is

�i� �1� Fi�(�i�)fi�(�i�)

� 1� Fj(�j)fj(�j)

:

Finally, if we allocate to a bidder other than i� or j, the virtual surplus is thesame as from allocating to i�. Thus, misallocation could only increase infor-mation rents: allocating to a bidder other than i� still concedes informationrents to higher types of i� (who can buy the good cheaply in resale), and mayleave information rents to the ine¢ cient winner as well. So the seller cannotdo better than allocating to the high-value bidder i�, which leaves rents tohim only. Furthermore, we should allocate to the highest-value bidder i� ifand only if his virtual value is positive, which is equivalent to �i > ri = ��1i (0)provided that his virtual value function �i crosses zero just once (which isensured under assumption (A1)). In particular, if there are two bidders andeach �i crosses zero once, the allocation rule solving the relaxed problem isas shown in Figure 1, and the solution is unique up to measure-zero sets.Given the allocation rule, transfers consistent with (11) can be obtained,

in particular, by charging auction winner i� the threshold pricemaxfri� ; �(2)g,which is his lowest value for which he would have won the good given thevalues of others; and charging losers nothing. The resulting mechanism isknown as the Vickrey auction with lazy reserves (Dhangwatnotai et al.,2010), and it is dominant-strategy incentive compatible in the absence ofresale.However, with our resale procedure v, the solution to the relaxed problem

violates global incentive compatibility constraints (6) unless all bidders havethe same optimal reserve ri. For example, consider the case of two bidderswith r1 > r2, and consider bidder 1 of type �1 2 (r2; r1) deviating to report�̂1 < �1, as illustrated with the horizontal arrow in Figure 1. This deviationa¤ects the auctions outcome when �2 2 (maxf�̂1; r2g; �1), and in these casesit changes the outcome from leaving the object unsold to giving it to bidder2, allowing 1 to protably buy it back in resale. Thus, in order to nd thecorrect solution, we need to consider non-local incentive constraints.14

14Note that we could deter all local misreports within distance " > 0 by perturbingthe relaxed solution to withhold the good whenever j�2 � �1j < ", but the perturbedmechanism would still be vulnerable to global deviations of the form described above.This is in contrast to ironing in the standard screening setting, where global incentive

14

Figure 1: Allocation rule from relaxed problem. 1 means allocate to bidder1; 2 means allocate to bidder 2. In the remaining regions, the good is notsold.

4.3 Ausubel-Cramton-Vickrey Auctions

Intuitively, to avoid the incentive to underbid to cede the object and thenbuy it back, we might use an allocation rule in which a lower bid nevercauses the object to be sold. (Ausubel and Cramton (1999) call this propertymonotonicity in aggregate.) In the two-bidder example, we might try to xthe allocation rule by lling inthe triangular region r2 < �2 < �1 < r1 inFigure 1, allocating to bidder 1 in this region (based on the above intuitionthat we prefer to allocate to the high-value bidder or to nobody). Note,however, that this solution can be improved: since bidder 1s virtual valuein the lled-in triangle is negative, the seller would rather shrink the size ofthis triangle by raising the reserve price for bidder 2 above r2, even thoughdoing so also means missing out on protable sales to bidder 2. The optimalreserve price for bidder 2 trades o¤ these two e¤ects. The resulting allocationrule is shown in Figure 2.

compatibility is implied by local (rst- and second-order) incentive constraints (Archerand Kleinberg, 2014; Carroll, 2012).

15

Figure 2: Allocation rule for ACV auction. 1 means allocate to bidder 1; 2means allocate to bidder 2. In the remaining regions, the good is not sold.

This auction belongs to the following class of auctions, introduced byAusubel and Cramton (1999, 2004):

Denition 1. An Ausubel-Cramton-Vickrey (ACV) auction with reserveprices p1; : : : ; pn 2 [0; 1] is the direct revelation mechanism described as fol-lows:

� Allocation rule:

�i (�) =

�1 if i = i� (�) and �j > pj for some j;0 otherwise,

where i� (�) 2 argmaxi �i (with arbitrary tie-breaking.)

� Payments:

i (�) =

8 pj for some j 6= i;0 otherwise,

and non-participation message ; is treated the same as a report of 0.

16

In words, if at least one bidder i beats his reserve price pi, then the goodis allocated to the highest-value bidder, otherwise, the good is left unsold.Importantly, since the reserve prices are asymmetric, a bidder i can win thegood without meeting his reserve price pi if another bidder j with a lowerreserve has met his reserve pj.A winners payment in the ACV auction is his threshold price the

minimal bid that would have allowed him to win. By standard arguments,this ensures that the auction is strategy-proof without resale.15 More im-portant for us is that (in contrast to the solution to our relaxed problem)in an ACV auction, bidders are incentivized to bid truthfully even with re-sale. Specically, truthful bidding is an ex post equilibrium, for any resaleprocedure and for any prole of values:

Theorem 2. Consider an ACV auction with reserve prices p1; : : : ; pn. Forany resale payo¤ functions v1; : : : ; vn satisfying (1)(2), it is an ex post equi-librium for all bidders to participate and report their true values (after whichno resale occurs). Formally, for any values �1; : : : ; �n, and any possible de-viation �̂i 2 [0; 1] [ f;g for any bidder i,

�i�i(�)� i(�) � vi(�(�̂i; ��i); �)� i(�̂i; �):

The proof essentially follows the informal argument given in Section 3 forthe Vickrey auction without reserves, but also using the fact that in an ACVauction, a downward deviation would never cause the object to be sold. Whilethis result was proved by Ausubel and Cramton (2004), for completeness wegive a proof in Appendix C. Also, in that appendix we develop additionalformalism for a broader class of resale procedures that drops the assumptionthat values are revealed; thus, bargaining may take place under asymmetricinformation. The proof shows that the result holds for this broader class ofresale procedures.We note in passing that this theorem only gives us ex post equilibrium,

not dominant strategies as we would have in Vickrey auctions without re-15An ACV auction can also be indirectly implemented as a deferred acceptanceclock

auction of the kind described by Milgrom and Segal (2017). In this implementation, theauction o¤ers the same ascending price to all bidders, letting them either accept the priceor exit at any point, and stopping when both (i) there is a single bidder who is stillaccepting the current price, and (ii) at least one bidder (not necessarily the one who isstill bidding) has ever accepted a price above his reserve price. (Note, however, that oneadvantage of the clock auction format its obvious strategy-proofness (Li, 2017) doesnot hold when resale is possible.)

17

sale.16 This is to be expected since our setting is one of interdependent values(compare Perry and Reny (2002) or Chung and Ely (2006)).

4.4 Construction of Optimal Reserve Prices

In the two-bidder ACV auction illustrated in Figure 2, bidder 1s price p1can be optimized without regard to bidder 2 (since it only matters whenbidder 2 bids below p2 < p1, and therefore does not win), hence it is optimalto set p1 = r1. On the other hand, the optimal price for bidder 2 is raisedabove r2 to increase the expected revenue on bidder 1. Extending this ideato n bidders, we construct a weakly decreasing sequence p1; : : : ; pn of reserveprices and a corresponding sequence �R1; : : : �Rn of revenue levels on the rst kbidders recursively, initializing p0 = 1 and �R0 = 0 and letting for each k � 1,

�Rk = Rk (pk) + Fk (pk)��Rk�1 �Rk�1 (pk)

�(13)

= maxp2[0;pk�1]

�Rk (p) + Fk (p)

��Rk�1 �Rk�1 (p)

�; (14)

where Rk(p) is the expected revenue from the (symmetric) Vickrey auctionon the rst k bidders facing the same reserve price p, with R0(p) � 0.Formula (13) gives an inductive construction of the expected revenue �Rk

from the ACV auction on bidders 1; : : : ; k with reserves p1; : : : ; pk, assumingthe bidders bid their true values. To see this, compare this ACV auctionto the symmetric Vickrey auction with reserve pk on the same bidders. No-tice that the only states � in which the two auctions could yield di¤erentrevenues are those in which bidder ks value �k is below pk, which happenswith probability Fk(pk). Conditional on any such value of �k, the formerauction reduces to the ACV auction on the rst k � 1 bidders with reservesp1; : : : ; pk�1, yielding expected revenue �Rk�1, while the latter auction reducesto the Vickrey auction with reserve pk on the rst k � 1 bidders, yieldingexpected revenue Rk�1(pk).Condition (14) requires that bidder ks reserve price pk maximize rev-

enue on the strongest k bidders taking as given the stronger biddersreserves

16To see this concretely, imagine that there are two bidders, no reserve prices, and bidder1 expects 2 to bid higher than his true value. Then, under our resale procedure (8), 1 hasan incentive to underbid and make 2 win, since if 1 wins the object in the auction he hasto pay 2s (exaggerated) bid, whereas to buy it in resale he only has to pay 2s true value.Thus, truthful bidding is not a dominant strategy.

18

p1; : : : ; pk�1 and the constraint pk � pk�1. This condition does not immedi-ately imply full optimality of the reserve prices, since we do not know whetherthe constraint binds, or how the choice of pk in turn constrains the revenueson the weaker bidders. However, we will show a stronger result that theresulting auction is indeed optimal, and not just among ACV auctions butamong all possible auctions.17

4.5 Optimality of ACV Auction

We now come to our main theorem.

Theorem 3. Under assumptions (A1)-(A3), the formulas (13)-(14) uniquelydene reserve prices p1; : : : ; pn. The ACV auction with these reserve pricesis an optimal auction given resale procedure (8), and the resulting revenue is�Rn.

This implies a solution to our original problem:

Corollary 2. Under assumptions (A1)-(A3), the ACV auction with reserveprices p1; : : : ; pn uniquely dened by (13)-(14) solves the robust revenue max-imization problem (3).

We sketch the main steps of the proof of Theorem 3. Since Theorem 2showed that the ACV auction is feasible (i.e. satisifes the constraints (6)(7)), and its revenue is indeed �Rn by (13), we focus on showing that �Rn isan upper bound for revenue in any auction.As discussed in Subsection 4.2, we need to make active use of the non-

local incentive constraints, which, using the formula (11) for transfers, can

17 The working paper of Ausubel and Cramton (1999) formulated the problem of op-timal auction design, assuming no misallocation and monotonicity in the aggregate, mo-tivating both properties informally by incentive compatibility under perfect resale. Theyalso claimed (without proof) that, in the two-bidder case, the problem is solved by anACV auction. In contrast, we derive no misallocation and monotonicity in aggregate asproperties of a solution to the sellers maxmin problem under regularity assumptions (A1)-(A3). As argued in the Introduction, the optimal auction need not have these propertieswhen resale is perfect but not of the worst-case form. Also, in Appendix G we show thatthe robust-revenue-maximizing auction need not have these properties without regularityassumptions.

19

be rewritten entirely in terms of the allocation rule:Z �i�̂i

E~��i [v0i(�(� i;

~��i); � i; ~��i)] d� i�E~��i [vi(�(�̂i; ~��i); �i; ~��i)�vi(�(�̂i; ~��i); �̂i; ~��i)] � 0:

(15)We account for these constraints by introducing Lagrange multipliers on

them. Since there is a double continuum of such constraints for each bidder i,indexed by (�i; �̂i), the Lagrange multipliers (weights) on them are describedwith some appropriately constructed nonnegative measure Mi on [0; 1] �[0; 1]. The Lagrangian adds terms to the objective function (12) to penalizeviolations of the constraints:

E~�

"Xi

vi(�(~�); ~�)�

Xi

1� Fi(~�i)fi(~�i)

v0i(�(~�); ~�)

#+Xi

ZZSi(�i; �̂i;�) dMi(�i; �̂i);

(16)where Si(�i; �̂i;�) is the left side of (15) the slack in bidder is incentiveconstraint from type �i to type �̂i.We show Theorem 3 by constructing measuresM = (M1; : : : ;Mn) such

that h��;Mi satises the following conditions, where �� is the allocation rulefrom the ACV auction that is our candidate optimum.

(a) Optimization: Allocation rule �� maximizes Lagrangian (16) given mea-suresM;

(b) Complementary slackness: the support ofM is conned to constraints(15) that hold with equality at ��, i.e., Si(�i; �̂i; ��) = 0.

To see that these conditions imply the result, note that since M is anonnegative measure, the expected revenue from any incentive-compatibleand individually rational direct auction with allocation rule � cannot exceedthe value of the Lagrangian (16) at h�;Mi, which by (a) does not exceed thevalue of the Lagrangian at h��;Mi, which by (b) equals the expected revenuefrom ��.We construct Lagrange multiplier measures Mi to penalize deviations

that would be protable in the Vickrey auction with lazy reserves p1 �: : : � pn, of the kind that emerged as a solution to the relaxed problem.In this auction, as we argued, bidder i < n with value between pn and picannot win the object outright, and so would want to underbid to increasethe probability that the object is sold to another bidder, from whom he can

20

then buy in resale. Thus, we let the support of Mi(�i; �̂i) for bidder i < nbe the trapezoid described by pn � �i � pi and �̂i � �i, and setMn � 0 forthe weakest bidder. This ensures complementary slackness in the candidate-solution ACV auction: Bidder is deviation in the support can only a¤ectthe auctions outcome if i wins under truthtelling but his deviation cedes thegood to the next-highest bidder. Then, after the deviation, i buys in resaleat price �(2), which is the same price at which he would have won the auctionby bidding truthfully. So, i is indi¤erent between truthful bidding and thedeviation.We specically aim to construct measuresMi of the following form: for

appropriately chosen one-dimensional measures �i and �̂i with supports[pn; pi] and [0; pi] respectively, take their product (a two-dimensional mea-sure with support [pn; pi] � [0; pi]), and then restrict to the lower half-plane�̂i � �i; this restriction is Mi. We denote the two component measuresdistribution functions by �i(�i) and �̂i(�̂i), respectively.We then construct distribution functions �i and �̂i to satisfy the La-

grangian maximization condition (a). Once these are constructed, we verify(a) by expressing Lagrangian (16), which is a linear functional of the alloca-tion rule �, in the form

E~�

"Xi

�i(~�)�i(

~�)

#: (17)

The coe¢ cient �i(�), which we refer to as the modied virtual value ofbidder i, combines the ordinary virtual value and the terms coming fromincentive compatibility constraints (15) weighted by the measures M. Forthe measuresM we construct, we derive an explicit formula for �i(�) in theappendix (see (31)). Condition (a) then amounts to requiring that with prob-ability 1, the candidate allocation rule �� allocate the object to a bidder withthe highest modied virtual value provided it is positive, and not allocate atall if all bidders have negative modied virtual values.To illustrate how condition (a) serves to pin down measures �i and �̂i,

focus for simplicity on the two-bidder case, where we only need to construct ameasure for bidder 1. In order for the proposed ACV auction to be optimal,we need two properties to be satised:

1. When �1 < p2 and �2 = p2, bidder 2s modied virtual value should be0.

21

2. When �1 2 (p2; p1) and �2 = p2, bidder 1s modied virtual value shouldbe 0.

Each property is needed for condition (a) because a slight increase in �2should lead the auctioneer to sell the good to the highest-value bidder (whosemodied virtual value must then be positive), while a slight decrease in �2should lead the auctioneer to withhold the good (making the modied virtualvalue negative).Property #1 pins down the density of �̂1 on [0; p2]. It turns out to be

proportional to f1(�1) (we normalize the proportionality factor to 1). Thisoccurs because selling to bidder 2 in any such state (�1; p2) has two e¤ectson the Lagrangian: the direct e¤ect on revenue �2(�2), and the e¤ect oftightening the binding non-local incentive constraints of types of bidder 1above p2, who could deviate to �1 and then buy the good in resale. The rste¤ect appears with a weight proportional to the probability of state (�1; p2),while the second has weight proportional to �̂1(�1); these two weights mustbe proportional in order for these terms to cancel. Note that Property #1does not pin down the density of �̂1 on the rest of its domain, (p2; p1], but anatural guess is to take the density as f1(�1) here as well.Property #2 requires that the weighted slack in non-local incentive con-

straints of bidder 1s types above �1 induced by selling to �1 2 (p2; p1) exactlyo¤set bidder 1s negative virtual value �1 (�1). Once �̂1 is xed, this conditionpins down measure �1 on its domain [p2; p1].With more than two bidders, we need to similarly construct measures for

each bidder i < n. We again set �̂i to have the same density as is typedistribution, fi, restricted to the support [0; pi]. As for measures �i, they arepinned down by requiring the modied virtual value of the highest bidderi� to be exactly zero so that selling to him could be made contingent onwhether some other bidder j has beaten his reserve price, similarly to theargument above. The explicit formula ((30) in the appendix) depends on anauxiliary function H, constructed below. We show that under assumptions(A1)(A3), the constructed functions �i are increasing, so that eachMi isin fact a nonnegative measure.Finally, we verify that with these measures, condition (a) holds, i.e., the

ACV auction maximizes the modied virtual surplus (expressed as (17)) atevery type prole � without ties, and not just on the regions used in theindi¤erence conditions nailing down the measures. That is, we show thatthe highest-value bidder always has the highest modied virtual value, and

22

it is nonnegative when some bidders value is above his reserve price andnonpositive otherwise.Now, for the Corollary, letting �� be the described ACV auction, Theorem

3 establishes bound (4) with �R = �Rn, while bound (5) follows from Theorem2. Thus, by Lemma 1, �� solves the robust revenue maximization problem(and Bertrand resale procedure v solves the worst-case resale problem).

4.6 Calculation of Optimal Reserves

We now indicate briey how one might calculate the reserve prices pk in anapplication.An interior solution pk to maximization problem (14) must satisfy the

following rst-order condition:

�k (pk)Y

j

In Lemma 2 in Appendix D we show that function H is continuous,di¤erentiable, and satises the following di¤erential equation:

d

dp(H (p)�j

5.1 Robustness to Information Revelation

One might be concerned with an asymmetry in our model: we have allowedinformation to be revealed exogenously after the auction, but assumed it isnot revealed before the auction.A common perspective on optimal auction design is that ideally it should

not make assumptions on bidders information before the auction either.Indeed, without resale, Myersons (1981) optimal auction, though derivedunder the assumption that bidders share the auctioneers prior, can actuallybe implemented in dominant strategies, making it robust to biddersbeliefsabout each other. Similarly, in our setting with resale, we could imagine aseller who is concerned about bidders learning about each other both beforeand after the auction, and desires robustness to information arising at eitherstage. We have shown that an ACV auction is optimal when bidders sharethe designers prior before the auction and learn each others values afterit, following the worst-case resale procedure. But since truthful bidding isan ex post equilibrium in the ACV auction, it satises this stronger form ofrobustness. Thus, we get robustness to pre-auction information revelationfor free,without needing to require it explicitly in the sellers problem.One could alternatively try to restore symmetry to the model by assuming

that exogenous information revelation cannot happen either before or afterthe auction. In this case, our basic conclusion that robustness to resaleimplies the auctioneer should not misallocate no longer holds. For anexample, it su¢ ces to consider the must-sell case, where we need not worryabout (A1)(A3). Suppose that there are two bidders, with bidder 1s valueequal to 1 for sure, and bidder 2s value being 1=3 or 2=3 with probability 1=2each. Consider the mechanism that o¤ers the object to bidder 1 at a xedprice p = 2=3 � ", and if he rejects, gives it to bidder 2 for free. Note thatif bidder 1 rejects, he does not learn anything about bidder 2s value, so thehighest expected payo¤ he could hope to get in resale is 1=3 (which he couldobtain if he has all the bargaining power, by o¤ering either a price of 1=3or 2=3 to bidder 2). Thus, bidder 1 would prefer to buy in the mechanism.This gives a revenue of 2=3�", which exceeds the Vickrey auctions expectedrevenue of 1=2. This example involves discrete distributions, but it can easilybe perturbed to a continuous distribution with full support on [0; 1], and thenany e¢ cient must-sell auction is revenue-equivalent to Vickrey; and still, theseller would prefer our alternative auction over Vickrey, regardless of theresale procedure. So the optimal must-sell auction must misallocate with

25

positive probability, contradicting the conclusion of Theorem 1.18

In summary, our main conclusion that robustness to resale calls forno misallocation depends on allowing information to be revealed after theauction, but does not depend on whether information can be revealed beforethe auction.

5.2 Robustness to Value Distributions

Another seeming asymmetry arises between the modeling assumptions ofBayesian priors over bidders values and complete ignorance of the resaleprocedure. We may instead consider a more symmetric model in which foreach bidder i, there is a family of possible distributions Fi � �([0; 1]), ratherthan a single distribution, and the designers robust revenue maximizationproblem is

max�

�inf

v;F12F1;:::;Fn2FnRev(�; v; F1; : : : ; Fn)

�where Rev(�; v; F1; : : : ; Fn) stands for the supremum expected revenue in aBayes-Nash equilibrium of mechanism � with resale procedure v and distri-butions F1; : : : ; Fn.19 For this problem, our result about optimality of ACVauctions carries through. Specically:

Proposition 1. Assume that for each i, there is a lowest distributionFi 2 Fi that is rst-order stochastically dominated by every other F 0i 2 Fi.Assume that F1; : : : ; Fn have continuous densities and satisfy assumptions(A1)(A3). Then the robust revenue maximization problem is solved by theACV auction with reserves p1; : : : ; pn constructed in (13)-(14).

We prove the proposition by showing that the lowest distributions com-bined with resale procedure (8) constitute a worst case for the designer.

18Solving for the optimal auction in worst case over all resale procedures without in-formation revelation is a di¢ cult open question. We do know from Calzolari and Pavan(2006) that Myersons optimal biased allocation is generically not implementable even fora given resale procedure.19An alternative way to avoid the asymmetry would be by interpreting our analysis as

applying to a fully Bayesian auctioneer who believes that resale takes the particular worst-case form v, joining the literature on auction design for a given known resale procedure(cited in the Introduction). However, our preferred interpretation is that the auctioneerdoes not presume any given specic resale procedure, instead being completely ignorantabout it.

26

For this we need to verify that the optimal ACV auction for distributionsF1; :::; Fn would not yield lower expected revenue when biddersvalues aredrawn from higher distributions. While this kind of revenue monotonicitydoes not hold for an arbitrary ACV auction, we show that it holds for theoptimal ACV auction given distributions F1; :::; Fn.The above result still requires the designer to know lower bounds on the

possible distributions in order to choose the reserves. Can we say anythingwith no knowledge of the distributions at all? Our maxmin problem be-comes uninteresting: the worst case is that each buyer just has value 0 withprobability 1, so there is no positive revenue guarantee. However, we cangive two arguments to suggest that optimization of reserve prices has littlebenet when there are su¢ ciently many bidders, and in this case the Vick-rey auction with no reserves is a reasonable prior-free choice despite possibleasymmetries among bidders.20

First, we show that under uncertainty about resale, setting optimal re-serve prices is less valuable than adding just one bidder who is at least asstrong as the existing bidders:

Proposition 2. Suppose the value distributions of the n bidders are regu-lar. Under our worst-case resale procedure, any auction has a lower expectedrevenue than the Vickrey auction with one added bidder n+ 1, provided thatthe added bidders value distribution rst-order stochastically dominates thedistributions of all other bidders.

The proposition extends the classic result of Bulow and Klemperer (1996)to asymmetric settings with uncertainty about resale. In contrast, as notedby Hartline and Roughgarden (2009), the result does not extend to asymmet-ric settings without resale, in which Myersons optimal biased auction couldyield substantially higher revenue than the Vickrey auction with duplicatebidders.21

Second, we show that the di¤erence in revenue between the optimal auc-tion under worst-case resale and the Vickrey auction is very small when thenumber of bidders n is large: it shrinks to zero exponentially fast as n grows.

20In the propositions below we do not impose assumptions (A1)-(A3), because they areproved by showing that Vickrey does well even compared to the solution of the relaxedproblem, which relied only on regularity (a weakening of (A1)).21Namely, in their two-bidder Example 4.6, adding a third bidder with value c.d.f.

F3 (�) = min fF1 (�) ; F2 (�)g and using the Vickrey auction on the three bidders wouldonly yield 34 of the revenue of the optimal biased auction on the original two bidders.

27

(We require value distributions to satisfy a particular bound uniformly inn, e.g., they could be drawn from some nite family.) In contrast, if resalewere impossible, Myersons optimal auction could attain an improvementover Vickrey that shrinks only polynomially in n (even if just two di¤erentvalue distributions are possible). This suggests that uncertainty about re-sale is quantitatively more important for auction design than optimizationof reserve prices.

Proposition 3.

(a) If all distributions are regular and Fi(rj) � c < 1 for all i; j, then thedi¤erence between the expected revenue from the optimal auction underworst-case resale and the expected revenue from the Vickrey auction isat most n � cn�1.

(b) Let F , F̂ be regular distributions with respective twice-di¤erentiable den-sities f , f̂ such that f (1) ; f̂ (1) > 0 and f 0 (1) =f (1) < f̂ 0 (1) =f̂ (1).There is a constant c0 > 0 with the following property: if there are n=2bidders drawn from distribution F and n=2 from distribution F̂ , theexpected revenue from the optimal Myerson auction (without resale)exceeds that of the Vickrey auction by at least c0=n3.

5.3 Relaxing Regularity

Our regularity assumptions (A1)-(A3) are not necessary for ACV auctionsto be optimal. For example, without these assumptions we can state22

Proposition 4. If all biddersvirtual value functions �i(�i) cross zero justonce and all at the same point r, the Vickrey auction with symmetric reserveprice r solves the robust revenue maximization problem.

Indeed, this auction solves the relaxed problem in Section 4.2. From this(combined with Theorem 2), the proposition follows immediately.

22For another example, in the two-bidder case, assumption (A3) is not needed at all(it su¢ ces to assume that r1 > r2), while assumptions (A1)-(A2) could be replaced withthe non-nested assumptions that the function R1(p) = p(1 � F1(p)) is concave, and that�2 is increasing. (The proof for this case necessitates constructing a di¤erent Lagrangemultiplier measure than the one we use.)

28

In particular, while we have focused on settings where the Myerson opti-mal auction is ine¢ cient due to bias, it can also be ine¢ cient due to random-ization: when bidders are symmetric but have non-monotone virtual valuefunctions, the Myerson auction calls for random allocation when multiplebidders are in an ironing region. The above proposition shows that in thesecases, too, robustness to resale can make it optimal to allocate e¢ ciently.However, we cannot dispense with distributional assumptions entirely:

Appendix G describes an example with symmetric bidders where the optimalauction is not of the ACV form. We do not know of any simple, generalassumptions that nest all known cases where ACV auctions are optimal.

5.4 Further possibilities

One might argue, as Börgers (2017) does, that the maxmin criterion used in(3) is too permissive. Applied to our setting, his critique would say that theACV mechanism we have identied as maxmin optimal is weakly dominated,in the sense that there are other mechanisms that do at least as well forevery resale procedure, and strictly better for some. For example, if theresale procedure is common knowledge among bidders, the auctioneer canelicit it from them and then run an optimal auction tailored to that resaleprocedure. However, this elicitation mechanism would not work if the resaleprocedure is not common knowledge, while an ACV mechanism would stillremain maxmin-optimal.23

While we have assumed the worst case with respect to resale procedures,we have nonetheless focused on the best equilibrium, implicitly assuming thatthe seller gets to choose the equilibrium played by the bidders (as is usualin mechanism design). However, it is known that Vickrey auctions can havemany equilibria when resale is possible, including ones that are ine¢ cientand have lower revenue than the truthful equilibrium but are preferred by allbidders (Garratt and Tröger, 2006; Garratt, Tröger, and Zheng, 2009). Weexpect the same for ACV auctions. It is an open question whether the ACVauction can be modied in a way that makes its expected revenue guaranteerobust to equilibrium selection.24

23As Börgers (2017) notes, an undominated mechanism may not exist in this case, sincein addition to any mechanism the seller could run a side betting mechanism collectingcommissions on bets when bidders have heterogenous beliefs.24In the absence of resale, it is possible to acheve robustness to equilibrium selection,

as well as stronger forms of bidder collusion: La¤ont and Martimort (1997) and Che and

29

Finally, we have presented here a model in which resale can occur onlybecause the outcome of the auction was ine¢ cient. In practice, however,resale can occur for many reasons, and it would be natural to consider robustauction design for situations where resale is socially desirable: for example,buyers are uncertain about their own values and will learn more about themafter the auction (as in Haile, 2003). Presumably the optimal auctions insuch models would involve resale occurring in equilibrium.

6 Summary

We began from the classic theory of optimal auctions, which suggests thatit can be optimal to bias the auction in favor of weaker bidders. A roughintuition suggests that such misallocation is no longer advantageous whenit can be undone by resale among the bidders. A series of simple examplesshows, however, that when the auctioneer knows the resale procedure, ittypically is still optimal to misallocate, even if resale is perfect(contraryto the intuition of Ausubel and Cramton (1999)).We provided a model in which resale makes it optimal not to misallocate.

In our model, the auctioneer does not know the resale procedure and desiresrevenue to be robust to this uncertainty. In the must-sell version of themodel, the optimal auction is simply a Vickrey auction. In the can-keepversion, the optimum (under appropriate regularity conditions) is an ACVauction: The auctioneer boosts revenue by setting reserve prices, as in theclassic model, but never benets from misallocating the good.

References

[1] Archer, A., and R. Kleinberg (2014), Truthful Germs are Contagious:A Local-to-Global Characterization of Truthfulness,Games and Eco-nomic Behavior 86, 340366.

[2] Ausubel, L. M., and P. Cramton (1999), The Optimality of Being Ef-cient,working paper, Department of Economics, University of Mary-land, College Park.

Kim (2006) show how this can be done with any Bayesian incentive-compatible auction,including Myersons (1981) optimal auction. Che and Kim (2006, footnote 34) explainwhy their results on collusion do not also imply robustness to resale.

30

[3] Ausubel, L. M., and P. Cramton (2004), Vickrey Auctions with ReservePricing,Economic Theory 23(3), 493-505.

[4] Bagnoli, M., and T. Bergstrom (2005), Log-Concave Probability andIts Applications,Economic Theory 26(2), 445469.

[5] Bergemann, D., B. Brooks, and S. Morris (2017a), Selling to Interme-diaries: Optimal Auction Design in a Common Value Model,workingpaper, University of Chicago.

[6] Bergemann, D., B. Brooks, and S. Morris (2017b), InformationallyRobust Optimal Auction Design,working paper, University of Chicago.

[7] Börgers, T. (2017), (No) Foundations of Dominant-Strategy Mecha-nisms: A Comment on Chung and Ely (2007), Review of EconomicDesign 21, 7382.

[8] Bulow, J., and P. Klemperer (1996), Auctions Versus Negotiations,American Economic Review 86(1), 180-194.

[9] Bulow, J., and P. Klemperer (2002), Prices and the Winners Curse,RAND Journal of Economics 33(1), 1-21.

[10] Calzolari, G., and A. Pavan (2006), Monopoly with Resale,RANDJournal of Economics 37(2), 362375.

[11] Carroll, G. (2012), When are Local Incentive Constraints Su¢ cient?Econometrica 80(2), 661686.

[12] Carroll, G. (2015), Robustness and Linear Contracts,American Eco-nomic Review, 105(2), 536-563.

[13] Che, Y.-K., and J. Kim (2006), Robustly Collusion-Proof Implementa-tion,Econometrica 74(4), 10631107.

[14] Chung, K.-S., and J.C. Ely (2006), Ex-Post Incentive CompatibleMechanism Design,working paper, Northwestern University.

[15] Chung, K.-S., and J.C. Ely (2007), Foundations of Dominant StrategyMechanisms,Review of Economic Studies 74, 447476.

31

[16] Daskalakis, C., A. Deckelbaum, and C. Tzamos (2013), MechanismDesign via Optimal Transport,in Proceedings of the 14th Annual ACMConference on Economics and Computation (EC 13), 269286.

[17] Dhangwatnotai, P., T. Roughgarden, and Q. Yan (2010), RevenueMax-imization with a Single Sample,in Proceedings of the 11th Annual ACMConference on Economics and Computation (EC 10), 129138.

[18] Falk, M., J. Hüsler, and R.-D. Reiss (2010), Laws of Small Numbers:Extremes and Rare Events (third edition). Birkhäuser.

[19] Frankel, A. (2014), Aligned Delegation,American Economic Review,104(1), 66-83.

[20] Garratt, R., and T. Tröger (2006), Speculation in Standard Auctionswith Resale,Econometrica 74(3), 753769.

[21] Garratt, R., T. Tröger, and C. Z. Zheng (2009), Collusion via Resale,Econometrica 77(4), 10951136.

[22] Haile, P. (2003), Auctions with Private Uncertainty and Resale Oppor-tunities,Journal of Economic Theory 108 (1), 72110.

[23] Hartline, J., and T. Roughgarden (2009), Simple versus Optimal Mech-anisms,in Proceedings of the 10th ACM conference on Electronic Com-merce (EC 09), 225234.

[24] Jehiel, P., and B. Moldovanu (2000), Auctions with Downstream Inter-action among Buyers,RAND Journal of Economics 31 (4), 768-791.

[25] La¤ont, J.-J., and D. Martimort (1997), Collusion under AsymmetricInformation,Econometrica 65, 875911.

[26] Li, S. (2017), Obviously Strategy-Proof Mechanisms,American Eco-nomic Review 107 (11), 32573287.

[27] McAfee, R.P., and J. McMillan (1989), Government Procurement andInternational Trade,Journal of International Economics 26, 921943.

[28] Milgrom, P., and I. Segal (2002), Envelope Theorems for ArbitraryChoice Sets,Econometrica 70(2), 583-601.

32

[29] Milgrom, P., and I. Segal (2017), Deferred-Acceptance Auctions andRadio Spectrum Reallocation,working paper, Stanford University.

[30] Milgrom, P., and R. Weber (1985), Distributional Strategies forGames with Incomplete Information,Mathematics of Operations Re-search 10(4), 619632.

[31] Myerson, R. (1981), Optimal Auction Design,Mathematics of Opera-tions Research 6(1), 58-73.

[32] Myerson, R., and P. Reny (2015), Open Sequential Equilibria of Multi-Stage Games with Innite Sets of Types and Actions,working paper,University of Chicago.

[33] Myerson, R., and M. Satterthwaite (1983), E¢ cient Mechanisms forBilateral Trading,Journal of Economic Theory 29, 265-281.

[34] Perry, M., and P. Reny (2002), An E¢ cient Auction,Econometrica70(3), 11991212.

[35] Tirole, J. (1988), The Theory of Industrial Organization. MIT Press.

[36] Yamashita, T. (2015), Implementation in Weakly Undominated Strate-gies: Optimality of Second-Price Auction and Posted-Price Mechanism,Review of Economic Studies 82, 1223-1246.

[37] Yamashita, T., and S. Zhu (2017), On the Foundations of Ex PostIncentive Compatible Mechanisms,working paper, Toulouse School ofEconomics.

[38] Zheng, C.Z. (2002), Optimal Auctions with Resale, Econometrica70(6), 21972224.

33

A Proof of Lemma 1

(4)-(5) imply that for all auctions �, infv Rev(�; v) � �R � infv Rev(��; v),hence �� solves the robust revenue maximization problem, and the problemsvalue is �R. Also, (4)-(5) imply that for all resale procedures v, sup�Rev(�; v) ��R � sup�Rev(�; v), hence v solves the worst-case resale problem, and theproblems value is �R. Plugging in � = �� and v = v into (4)-(5) yieldsRev(��; v) = �R.

B Proof of Theorem 1

Take any direct auction mechanism h�; i satisfying (6)-(7), and dene in-terim expected payo¤s Ui(�i) as in (9). Rewrite incentive compatibility (6)as

Ui (�i)�Ui(�̂i) � E~��i [vi(�(�̂i; ~��i); �i; ~��i)�vi(�(�̂i; ~��i); �̂i; ~��i)] for all �i; �̂i:(23)

SinceP

j �j(~�) = 1 with probability 1, there exists arbitrarily small �̂i such

thatP

j �j(�̂i;~��i) = 1 with probability 1. Note that for any prole �

such that minj �j > �̂i and x � �(�̂i; ��i) satisesP

j xj = 1, we have

vi(x; �) � vi(x; �̂i; ��i) � �i � �(2) if �i > �(2) and vi(x; �) � vi(x; �̂i; ��i)otherwise, hence vi(x; �) � vi(x; �̂i; ��i) � max

��i � �(2); 0

. For any other

�,���vi(x; �)� vi(x; �̂i; ��i)�max��i � �(2); 0��� � 2. Hence, taking expecta-

tions, (23) implies

E~�ihUi(~�i)

i� Ui(�̂i) � E~�

hmax

n~�i � ~�(2); 0

oi� 2

�1�

Yj

h1� Fj(�̂i)

i�:

Since the inequality has to hold for arbitrarily small �̂i, and Ui(�̂i) � 0by individual rationality (7), we must have

E~�ihUi(~�i)

i� E~�

hmax

n~�i � ~�(2); 0

oi:

Therefore,

E~�

"Xi

i(~�)

#= E~�

"Xi

vi(�(~�); ~�)

#�Xi

E~�ihUi(~�i)

i� E~�

h~�(1)

i� E~�

"Xi

maxn~�i � ~�(2); 0

o#= E~�

h~�(2)

i,

34

which is the expected revenue in the Vickrey auction.

C Resale-Proofness of ACV Auctions

In this appendix we prove Theorem 2, and in the process we formulate ageneralization that applies to a broader class of resale procedures, which maytake place under asymmetric information. For this purpose, we assume theauction has the ACV form, and we allow the outcome of the resale bargainingto depend on biddersreports �̂1; : : : ; �̂n in the auction, since these reportsmay a¤ect biddersbeliefs about each others values and consequently theirbehavior in bargaining. Therefore, we now describe the payo¤ that bidder ireceives in post-auction bargaining by a function

vi(x; �̂1; : : : ; �̂n; �1; : : : ; �n);

where �1; : : : ; �n are the bidderstrue values, �̂1; : : : ; �̂n are the values thatwere reported in the auction, and x was the allocation chosen by the auc-tion.25 ;26 This formalism lets us distinguish the dependence of is payo¤on �̂j(which can inuence the biddersbeliefs about js type at the time bargain-ing begins) from the dependence on �j (which can inuence how j actuallybehaves in bargaining).27

25For simplicity, we assume that after the ACV auction is completed, the auctioneerdiscloses all bids. This avoids the need to model a bidders inference about bids fromimperfect information disclosed in the auction (e.g., his own allocation).26Note that we continue modeling resale using reduced-form payo¤s rather than as an

explicit noncooperative game following the auction game. The problem with the latterapproach is that to ensure individual rationality in a nonncooperative resale game, eachbidder should have access to a non-participationaction. However, then our formulationof the robust revenue maximization problem is too weak: any auction that is incentive-compatible without resale in particular the Myerson (1981) optimal auction hasan equilibrium in which every agent always bids truthfully and then never participates inresale. We could try to rule these out by restricting to equilibria satisfying some perfectionrequirement in the denition of Rev(�; v). But we want to ensure that a reasonably broadclass of mechanisms can guarantee equilibrium existence for all v, and existence of perfectequilibria with innite type spaces is a thorny problem (see Myerson and Reny, 2015).27We comment that our approach to modeling resale payo¤s here relies on having xed

the mechanism and the proposed equilibrium. In a general mechanism, if we did notassume full information in resale, then the e¤ect of messages on resale bargaining woulddepend on the o¤-path beliefs they induce within the equilibrium being played; thus wewould have a circularity, with resale payo¤s depending on the choice of equilibrium and

35

So a resale procedure is described by a prole of functions v1; : : : ; vn :X � [0; 1]2n ! R, satisfying the conditions that total resale payo¤s do notexceed the total surplus available among the bidders:X

i

vi(x; �̂; �) � (maxi�i) �

Xi

xi

!(24)

for all x; �̂; �; and are individually rational :

vi(x; �̂; �) � �ixi (25)

for all i,x; �̂; �.28

Theorem 4. Consider an ACV auction with reserve prices p1; : : : ; pn. Forany resale payo¤ functions v1; : : : ; vn satisfying (24)(25), it is an ex postequilibrium for all bidders to participate and report their true values (afterwhich no resale occurs). Formally, for any values �1; : : : ; �n, and any possibledeviation �̂i 2 [0; 1] [ f;g for any bidder i,

�i�i(�)� i(�) � vi(�(�̂i; ��i); �̂i; ��i; �)� i(�̂i; ��i): (26)

We now give a proof of the theorem. Note that we need not considerthe non-participation message �̂i = ; separately, since it is treated by themechanism as equivalent to reporting 0. We consider all possible cases.

� Suppose that bidder i wins under truth-telling: �i(�) = 1. Then theleft-hand side of (26) is �i � i(�) � 0.

If i deviates to �̂i such that he still wins, then he pays the sameprice (and no resale occurs).

If i deviates to �̂i such that some bidder j 6= i wins (and i thenpays zero), this is only possible if �j = maxk 6=i �k = i(�) (isthreshold price under truthtelling). Then (25) implies that theresale payo¤ of bidder j is at least �j and the resale payo¤s ofall bidders k =2 fi; jg are nonnegative, and therefore by (24) theresale payo¤ of bidder i is at most �i � �j. So the right side of(26) is at most �i � �j which equals the left side.

vice versa. Since these modeling technicalities are not our focus, we have avoided them byfocusing on full-information resale for the formal statement of the robust revenue problem(3).28Note that (25) is an ex post individual rationality constraint.

36

If i deviates to �̂i such that the object goes unallocated, then noresale is possible, so the right side of (26) is zero.

� Suppose that some bidder j 6= i wins under truth-telling: �j(�) = 1.Then the left-hand side of (26) is zero.

If i deviates to a �̂i that does not win the object, then either jstill wins, or the object is unsold. Either way, no resale occurs,and the right side of (26) is zero.

If i deviates to win the object, he pays the threshold price i(�̂i; ��i) =�j = maxk �k. By (25), the resale payo¤s of all other bidders arenonnegative, and therefore by (24) the resale payo¤ of bidder iis at most maxk �k = �j. So the right side of (26) is at most�j � �j = 0.

� Suppose that the object is left unsold under truth-telling: �(�) = 0.Then the left-hand side of (26) is zero.

If i deviates such that the object remains unsold, then no resaleis possible and the right side of (26) is zero.

If i deviates such that some bidder j 6= i wins the object, thenallocating to j is e¢ cient for the reported value prole (�̂i; ��i),and �̂i is above the reserve while the true value �i is not:

�j � �̂i > pi � �i; and �j � �k for all k 6= i:

So allocating to j is e¢ cient for the true value prole �, and noresale is possible, so the right side of (26) is zero.

If i deviates to win the object, he pays the threshold price i(�̂i; ��i) =maxfpi;maxj 6=i �jg. By (25), the resale payo¤s of all other bid-ders are nonnegative, and therefore by (24) the resale payo¤ ofbidder i is at most maxj �j. Since �i � pi (because the objectis unsold under truth-telling), this expression does not exceedmaxfpi;maxj 6=i �jg = i(�̂i; ��i), so the right side of (26) is atmost zero.

37

D A Key Lemma

The following lemma is key to constructing the multipliers used to proveTheorem 3.

Lemma 2. Under assumptions (A1)-(A3), there exist unique sequences p1 �� � � � pn and �R1; : : : ; �Rn satisfying (13)-(14). The sequence of prices pk andthe function H : [pn; 1) ! R dened by (19) are jointly characterized by thefollowing two properties:

(i) pk > 0 and (20) holds for each k � 1,

(ii) H is di¤erentiable and satises di¤erential equation (21) and the bound-ary condition H(p) = 0 for p 2 [p1; 1).

Furthermore,

(iii) the function H is nonincreasing, and

(iv) the reserve prices satisfy pk � rk for each k (with equality for k = 1).

The lemma follows from the following four claims:

Claim 1. There exist reserve prices and revenue levels satisfying (13)-(14).

Claim 2. When reserve prices and revenue levels satisfy (13)-(14) and thefunction H is dened by (19), properties (i)-(ii) hold.

Claim 3. When properties (i)-(ii) hold, property (iii) also holds.

Claim 4. There is a unique price sequence p1; : : : ; pn and a correspondingfunction H such that properties (i)-(iii) hold. These prices satisfy (iv).

We prove these claims in turn.

D.1 Proof of Claim 1

The claim follows from the continuity of the objective function in (14).

38

D.2 Proof of Claim 2

First, note that H as dened by (19) is continuous at the endpoints of theintervals of its denition, i.e. the points p = pk. This follows from (13). (Itis immediate that H is continuous elsewhere.)We now show properties (i)-(ii) in turn:

(i) First we argue that the derivative of the expected revenue Rk(p) inthe Vickrey auction with symmetric reserve p is given by the followingformula:

R0k(p) = �kXi=1

0BB@ kYj=1j 6=i

Fj(p)

1CCA fi(p)�i(p): (27)Indeed, we can express the value of this Vickrey auction as the integralof virtual surplus, as in (12):

Rk(p) = E~�h1n~�i � p for some i

o� �i�(~�)(~�i�(~�))

i:

Equivalently, this is the integral of �i�(�)(�i�(�)) over the region wherebidder i�(�) has value at least p. As p increases marginally, the regionshrinks by losing the surface where one bidders value is p and all otherbidders values are below p. Therefore, the derivative of Rk is thenegative of the integral of virtual surplus over this surface. For eachbidder i, there is one portion of the surface where is value is p (andother bidders are below p), and the virtual surplus is �i(p).

Now, we show that property (i) holds, and in addition that �Rk > Rk (0),by induction on k � 1. This holds for k = 1, as the optimal price mustsatisfy p1 2 (0; 1) to achieve revenue �R1 > R1 (0) = 0, and must satisfythe rst-order condition �1 (p1) = H (p1) = 0. Now, suppose that theinductive hypothesis holds for k � 1. Using (27), express

R0k (p) = Fk (p)R0k�1 (p)�

k�1Yi=1

Fi (p)

!�k (p) fk (p) ;

and calculate the derivative of the objective function in (14) for p 2[0; pk�1] as

Dk (p) � R0k (p) + fk (p)��Rk�1 �Rk�1 (p)

�� Fk (p)R0k�1 (p)

= fk (p)��Rk�1 �Rk�1 (p)� �k (p)

Qk�1i=1 Fi (p)

�.

39

In particular, Dk (0) = fk (0)��Rk�1 �Rk�1 (0)

�> 0 by the inductive

hypothesis for k � 1, and therefore we have �Rk > Rk (0) and pk > 0.Now we can express

Dk (pk) = fk (pk) (H (pk)� �k (pk))Qk�1i=1 Fi (pk) ;

where H is given by (19). Consider two remaining cases:

If pk 2 (0; pk�1) then the necessary rst-order condition for maxi-mization (14) is Dk (pk) = 0, and therefore H (pk) = �k (pk).

If pk = pk�1 then the necessary rst-order condition for maximiza-tion (14) is Dk (pk) � 0, and therefore H (pk) � �k (pk). On theother hand, we have

H (pk) = H (pk�1) = �k�1 (pk�1) � �k (pk�1) = �k (pk) ;

where the second equality uses the inductive hypothesis and theinequality uses assumption (A3). Combined with the above, weagain have H (pk) = �k (pk) :

(ii) From (19) and the observation that H is continuous at the intervalendpoints, H(p)

Qj

Then the fact that the left-hand and right-hand derivatives agree at pkis immediate, since the right side of (28) on interval [pk+1; pk] di¤ersfrom the formula for interval [pk; pk�1] only by inclusion of the j = kterm, which is zero at pk by property (i).

D.3 Proof of Claim 3

Using equation (28) above and assumption (A3), we see that for any p 2[pk; pk�1] such that H (p) � �k�1 (p) we have H 0 (p) � 0, and the inequalityis strict if H (p) > �k�1 (p).Now, by property (i),H (pk�1) = �k�1 (pk�1), and thereforeH 0 (pk�1) � 0,

hence for p in a left-neighborhood of pk�1,

H (p)� �k�1 (p) � �k�1 (pk�1) + o (pk�1 � p)� �k�1 (p)� pk�1 � p+ o (pk�1 � p) > 0;

where the second inequality uses (A1). Thus, we can choose p̂ < pk�1 closeenough to pk�1 such that H(p) > �k�1 (p) for all p 2 [p̂; pk�1), and thereforeH 0 (p) < 0 on this interval. In particular this implies H(p̂) > H(pk�1). Butthen H (p) cannot cross �k�1 (pk�1) anywhere in the interval [pk; pk�1]: in-deed, otherwise, letting p� = max fp 2 [pk; p̂] : H (p) � �k�1 (pk�1)g we wouldhave

0 = H(pk�1)� �k�1 (pk�1) < H(p̂)� �k�1 (pk�1) =Z p̂p�H 0 (p) dp; (29)

while on the other hand we would have H(p) � �k�1 (pk�1) � �k�1 (p) for allp 2 (p�; p̂) and therefore H 0 (p) � 0 for all p 2 (p�; p̂), making the right-handside of (29) nonpositive a contradiction.Therefore, H(p) � �k�1(pk�1) � �k�1(p) for all p 2 [pk; pk�1], and there-

fore H 0 (p) � 0 on this interval. Since this holds for each k > 1, H isnonincreasing on [pn; p1].

D.4 Proof of Claim 4

We prove by induction on k � 1 that price pk is uniquely determined andsatises pk � rk. First, the claim holds for k = 1, since the boundarycondition H(p1) = 0 implies that (20) takes the form �1 (p1) = 0, and thisuniquely determines p1 = r1. Now, suppose that the claim holds for all j �

41

k�1, and in particular that prices p1; : : : ; pk�1 are uniquely determined. Thenby using property (ii) and integrating, the function H is uniquely determinedon the interval [pk; 1), whatever pk may be. Furthermore, by (i), pk must solve(20). Observe that:

� Any value of pk satisfying (20) must lie in pk 2 [rk; pk�1]. Indeed,suppose for contradiction that pk < rk. Then, using properties (i)and (iii), and strict monotonicity of �k (from (A1)), we have �k(pk) <�k(rk) = 0 = H(r1) � H(pk) = �k(pk), a contradiction.

� (20) has at most one solution on pk 2 [0; pk�1]. This holds because �kis strictly increasing by (A1) and H is nonincreasing by property (iii).

The two observations together imply that (20) has a unique solution pk 2[0; pk�1], which satises pk � rk. Thus, the inductive statement also holdsfor k.

E Proof of Theorem 3

The proof of Theorem 3 depends on Lemma 2 in Appendix D. Henceforthwe take this lemma, and the steps of its proof, as given.We also nd it useful to write vi(x; �) =

Pj vij(�)xj, where vij(�) =

maxf�i; �(2)g if i = j and = maxf0; �i � �(2)g otherwise. Write v0ij(�) for thederivative with respect to �i (which is dened almost everywhere).Now, to prove the theorem, for each i = 1; : : : ; n�1, dene one-dimensional

measures �i; �̂i, with supports [pn; pi] and [0; pi] respectively, whose distrib-ution functions are �̂i(�̂i) = Fi(�̂i) and �i(�i) = ��i(pn)� ��i(�i), where

��i(�i) =

8>:fi(pn)Fi(pn)

[H(pn)� �i(pn)] if �i < pn;fi(�i)Fi(�i)

[H(�i)� �i(�i)] if �i 2 [pn; pi];0 if �i > pi:

(30)

Note that by (A1) and Lemma 2(iii), the di¤erence H (�i) � �i(�i) isnonincreasing, and in particular H (�i)� �i(�i) � H(pi)� �i(pi) = 0. Usingin addition (A2), we see that ��i is nonincreasing, hence �i is nondecreasing,so it indeed describes a measure. Furthermore, by construction, ��i (�i), andso �i (�i), is constant on �i < pn and on �i > pi, hence �i is supported on[pn; pi].

42

Dene Mi(�i; �̂i) to be the restriction of the product measure �i � �̂ito the half-plane �̂i � �i. Also, dene ��n, Mn � 0. We already argued inSubsection 4.5 that the conditions (a)(b) stated there su¢ ce to prove thetheorem, and thatMi as dened here satises the complementary slacknesscondition (b), so the rest of the proof focuses on establishing the optimizationcondition (a).First we rewrite the Lagrangian in terms o